In statistics and information theory, the expected formation matrix of a likelihood function L ( θ ) {\displaystyle L(\theta )} is the matrix inverse of the Fisher information matrix of L ( θ ) {\displaystyle L(\theta )} , while the observed formation matrix of L ( θ ) {\displaystyle L(\theta )} is the inverse of the observed information matrix of L ( θ ) {\displaystyle L(\theta )} .

Currently, no notation for dealing with formation matrices is widely used, but in books and articles by Ole E. Barndorff-Nielsen and Peter McCullagh, the symbol j i j {\displaystyle j^{ij}} is used to denote the element of the i-th line and j-th column of the observed formation matrix. The geometric interpretation of the Fisher information matrix (metric) leads to a notation of g i j {\displaystyle g^{ij}} following the notation of the (contravariant) metric tensor in differential geometry. The Fisher information metric is denoted by g i j {\displaystyle g_{ij}} so that using Einstein notation we have g i k g k j = δ i j {\displaystyle g_{ik}g^{kj}=\delta _{i}^{j}} .

These matrices appear naturally in the asymptotic expansion of the distribution of many statistics related to the likelihood ratio.